Iterative methods for solving a class of monotone variational inequality problems with applications
- Haiyun Zhou^{1, 2}Email author,
- Yu Zhou^{2} and
- Guanghui Feng^{2}
https://doi.org/10.1186/s13660-015-0590-y
© Zhou et al.; licensee Springer. 2015
Received: 18 November 2014
Accepted: 5 February 2015
Published: 24 February 2015
Abstract
In this paper, we provide a more general regularization method for seeking a solution to a class of monotone variational inequalities in a real Hilbert space, where the regularizer is a hemicontinuous and strongly monotone operator. As a discretization of the regularization method, we propose an iterative method. We then prove that the proposed iterative method converges in norm to a solution of the class of monotone variational inequalities. We also apply our results to the constrained minimization problem and the minimum-norm fixed point problem for a generalized Lipschitz continuous and pseudocontractive mapping. The results presented in the paper improve and extend recent ones in the literature.
Keywords
MSC
1 Introduction
The monotone variational inequalities were initially investigated by Kinderlehrer and Stampacchia in [1] and have been widely studied by many authors ever since, due to the fact that they cover as diverse disciplines as partial differential equations, optimization, optimal control, mathematical programming, mechanics, and finance (see [1–23]).
It is well known that if F is k-Lipschitz continuous and η-strongly monotone, then MVIP (1.1) has a unique solution. Moreover, if one chooses \(\mu\in (0,\frac{2\eta}{k^{2}})\), then the sequence \(\{x_{n}\}\) generated by PGM (1.2) converges in norm to the unique solution to MVIP (1.1); see, for instance, [4, 8].
However, if F is simply k-Lipschitz continuous and monotone, but not η-strongly monotone, then MVIP (1.1) may fail to have a solution, which can be seen from the following counterexample.
Example 1.1
Let \(H=C=\mathbb{R}=(-\infty,\infty)\) and consider the function \(F:C\to H\) defined by \(F(x)=\arctan(x)-\frac{\pi }{2}\), \(x\in C\). Then F is 1-Lipschitz continuous and monotone. It is clear that the equation \(F(x)=0\) has no solutions in H, and hence MVIP (1.1) has no solutions.
For a k-Lipschitz continuous and monotone operator F, even though MVIP (1.1) has a solution, the PGM (1.2) associated with F does not yet necessarily converge to the solution of the MVIP (1.1).
Example 1.2
The Example 1.2 tells us that PGM (1.2) is invalid for a k-Lipschitz continuous and monotone operator F, and therefore further modifications to PGM (1.2) are needed. In this regard, we pick up the following several known results.
EM (1.3) contains two metric projections and it is indeed a composite of two classical projected gradient methods. We also remark in passing that in an infinite-dimensional Hilbert space, Korpelevich’s EM (1.3) has only weak convergence, in general, moreover, it cannot be used to seek the minimum-norm solution of MVIP (1.1).
Recently, Xu and Xu [22] provided a general regularization method for solving MVIP (1.1), where the regularizer is a Lipschitz continuous and strongly monotone operator. They also introduced an iterative method as discretization of the regularization method. They proved that both regularization and iterative methods converge in norm to a solution to MVIP (1.1) under some conditions.
Very recently, Iemoto et al. [23] studied a variational inequality for a hemicontinuous and monotone operator over the fixed point set of a strongly nonexpansive mapping in a real Hilbert space. They proposed an iterative algorithm and analyzed the weak convergence of the proposed algorithm.
On the other hand, the construction of fixed points for pseudocontractive mappings has been studied extensively by several authors since 1974. A good number of results was reported recently; see, for instance, [24–30].
Question
Can the Lipschitz continuity assumptions be removed or weakened in the results mentioned above?
The purpose of this paper is to answer the question mentioned above. In order to realize this objective, we first establish a new existence and uniqueness theorem for MVIP (1.1), where \(F:C\to H\) is a hemicontinuous and strongly monotone operator. By using the established existence and uniqueness theorem, we then introduce an implicit method for seeking a solution of MVIP (1.1). We also introduce an explicit iterative method. We prove that both the implicit and the explicit iterative methods converge in norm to the same solution of MVIP (1.1). Some applications are also included.
The rest of the paper is organized as follows. Section 2 contains some necessary concepts and useful facts. The new existence and uniqueness theorem for MVIP (1.1) and several convergence results of the proposed algorithms are established in Section 3. Finally, in Section 4, we provide some applications to the constrained minimization problem for a convex and continuously Fréchet differentiable functional and the minimum-norm fixed point problem for a generalized Lipschitz continuous and pseudocontractive mapping.
2 Preliminaries
Throughout this paper, we will assume that H is a real Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and its induced norm \(\| \cdot\|\). Let C be a nonempty, closed, and convex subset of H. We denote the strong convergence and weak convergence of \(\{x_{n}\}\) to \(x\in H\) by \(x_{n}\to x\) and \(x_{n}\rightharpoonup x\), respectively. We use ℝ to denote the set of real numbers. Let \(T:C\to H\) be a mapping. We use \(\operatorname{Fix}(T)\) to denote the set of fixed points of T. We also denote by \(\mathfrak{D}(T)\) and \(\mathfrak{R}(T)\) the domain and range of T, respectively. The letter I stands for the identity mapping on H.
In what follows, we shall collect some important concepts, facts, and tools, which will be used in Section 3.
Lemma 2.1
(see [11])
- (C1)Given \(x\in H\) and \(z\in C\).Then \(z=P_{C}x\) if and only if the inequalityholds.$$ \langle x-z,y-z\rangle\le0,\quad \forall y\in C $$(2.4)
- (C2)in particular, one has$$ \|P_{C}x-P_{C}y\|^{2}\le\langle P_{C}x-P_{C}y,x-y\rangle,\quad \forall x,y\in H, $$(2.5)$$ \|P_{C}x-P_{C}y\|\le\|x-y\|,\quad \forall x,y\in H. $$(2.6)
- (C3)in particular, one has$$ \bigl\| (I-P_{C})x-(I-P_{C})y \bigr\| ^{2}\le \bigl\langle (I-P_{C})x-(I-P_{C})y, x-y \bigr\rangle , \quad \forall x,y \in H, $$(2.7)$$ \bigl\| (I-P_{C})x-(I-P_{C})y \bigr\| \le\|x-y\|,\quad \forall x,y\in H. $$(2.8)
Recall that an operator \(F:C\to H\) is said to be
(v) demicontinuous if \(Fx_{n}\rightharpoonup Fx\) as \(n\to\infty\), whenever \(x_{n}\to x\) for any \(\{x_{n}\}\subset C\) and \(x\in C\);
(viii) an operator \(A\subset H\times H\) is called maximal monotone if it is monotone and it is not properly contained in any other monotone. We denote by \(\mathfrak{G}(A)\) the graph of A.
It is well known that T is pseudocontractive if and only if \(F=I-T\) is monotone.
Example 2.1
Let \(f:H\to\mathbb{R}\) be a convex and continuously Fréchet differentiable function. Then the gradient ∇f of f is maximal monotone and hemicontinuous.
It is clear that the following implication relation holds:
\(F:C\to H\) is k-Lipschitz continuous ⇒ F is continuous ⇒ F is demicontinuous ⇒ F is hemicontinuous, however, the converse relation does not hold true, which can be seen from the following counterexample.
Example 2.2
Consider a function \(\varphi:\mathbb{R}^{2}\to \mathbb{R}\) of two variables \(\varphi(x,y)=xy^{2}(x^{2}+y^{4})^{-1}\) for \((x,y)\in\mathbb{R}^{2}\backslash\{(0,0)\}\) and \(\varphi(0,0)=0\). Then φ is hemicontinuous but demicontinuous.
If \(F:\mathfrak{D}(F)=H\to H\) is monotone, then F is demicontinuous if and only if it is hemicontinuous; see, for instance, [9].
We remark that if an operator \(F:C\to H\) is either k-Lipschitz continuous or has a bounded range \(\mathfrak{R}(F)\), then it is generalized Lipschitz continuous. However, a generalized Lipschitz continuous mapping may be not continuous.
Example 2.3
Lemma 2.2
(see [7])
Lemma 2.3
(see [7])
Lemma 2.4
(see [7])
From Lemma 2.4, we know that \(VI(C,F)\) is closed convex, since \(T^{-1}0\) is closed convex.
Lemma 2.5
- (i)
\(\langle Fx,x - {x^{*}} \rangle\ge0\), \(\forall x \in C\);
- (ii)
\(\langle F{x^{*}},x - {x^{*}} \rangle\ge0\), \(\forall x \in C\).
From Lemma 2.5, we also deduce that \(VI(C,F)\) is closed convex.
Lemma 2.6
(see [31])
- (i)
\(\sum_{n = 0}^{\infty}{\gamma_{n}} = \infty\);
- (ii)
either \(\lim{\sup_{n \to\infty}}{\sigma_{n}} \le0\) or \(\sum_{n = 0}^{\infty}|{\gamma_{n}}{\sigma_{n}}| < \infty\).
Now we are in a proposition to state and prove the main results in this paper.
3 Main results
In this section we first establish a new existence and uniqueness theorem to MVIP (1.1) with F being a hemicontinuous and η-strongly monotone operator. We then introduce two kinds of algorithms (one implicit and the other explicit) for solving MVIP (1.1).
Theorem 3.1
Proof
From Theorem 3.1, we deduce immediately the following important result.
Corollary 3.1
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(F:C\to H\) be a hemicontinuous and η-strongly monotone operator. Then the MVIP (1.1) has a unique solution.
Theorem 3.2
Assume that \(VI(C,F)\neq\emptyset\). Then, the sequence \(\{y_{n}\}\) generated by (3.12) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u) \), in particular, if we take \(R=I\) and \(u=0\) in (3.12), then the sequence \(\{y_{n}\}\) generated by (3.12) converges in norm to the minimum-norm solution to MVIP (1.1).
Proof
Use \(y_{n}\rightharpoonup\hat{y}\) and (3.22) to conclude that \(y_{n}\to \hat{y}\) as \(n\to\infty\).
Theorem 3.1 tells us that \(\hat{y}\) is the unique solution of (3.24), which ensures that the whole sequence \(\{y_{n}\}\) converges in norm to \(\hat{y}\) as \(n\to \infty\). Moreover, it follows from Lemma 2.1(C1) and (3.24) that \(\hat {y}=P_{VI(C,F)}[(I-R)\hat{y}+u]\). This completes the proof. □
We next introduce an explicit iterative method for solving MVIP (1.1).
Theorem 3.3
- (i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{{\beta _{n}}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);
- (ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);
- (iii)
\(\frac{{|{\alpha_{n}} - {\alpha_{n - 1}}| + |{\beta_{n}} - {\beta _{n - 1}}|}}{{{\alpha_{n}}^{2}}} \to0\) as \(n \to\infty\).
Assume that \(VI(C,F)\neq\emptyset\). Then, the sequence \(\{x_{n}\}\) generated by (3.25) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u)\).
Proof
Let \(\{y_{n}\}\) be defined by (3.12). By using Theorem 3.2, we know that \(\{y_{n}\}\) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u) \). It is sufficient to show that \(x_{n+1}-y_{n}\to0\) as \(n\to\infty\). To end this, we first show that \(\{x_{n}\}\) is bounded.
By Lemma 2.6, we conclude that \(\Vert {{x_{n + 1}} - {y_{n}}} \Vert \to0\), as \(n \to\infty\), which means that \(\{ {x_{n}}\} \) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u)\). This completes the proof. □
Remark 3.1
Theorem 3.3 extends Theorem 3.1 of Xu and Xu [22] to the more general case, moreover, the choice of the iterative parameter sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) does not depend on the generalized Lipschitz constants of R and F.
Remark 3.2
4 Applications
In this section, we give some applications of the results established in Section 3.
By virtue of Theorem 3.3, we can deduce the following convergence result.
Theorem 4.1
Proof
Apply Theorem 3.3 to the case where \(F=\nabla\varphi\), \(R=I\), and \(u=0\) to get the conclusion. □
Finally, we apply our results to the minimum-norm fixed point problem for pseudocontractive mappings.
Theorem 4.2
- (i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{{\beta _{n}}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);
- (ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);
- (iii)
\(\frac{{|{\alpha_{n}} - {\alpha_{n - 1}}| + |{\beta_{n}} - {\beta _{n - 1}}|}}{{{\alpha_{n}}^{2}}} \to0\) as \(n \to\infty\).
Proof
Since \(T:C\to C\) is a generalized Lipschitz continuous, hemicontinuous, and pseudocontractive mapping, we deduce that F is a generalized Lipschitz continuous, hemecontinuous, and monotone operator. By Lemma 2.1(C1), noting that \(T:C\to C\) is a self-mapping, we see that \(VI(C,F)=\operatorname{Fix}(P_{C}T)=\operatorname{Fix}(T)\ne\emptyset\) by our assumption. Apply Theorem 3.3 to the case where \(F=I-T\) and \(R=I\) to derive the desired conclusion. □
Corollary 4.1
Let C be a nonempty, bounded, and closed convex subset of a real Hilbert space H. Let \(T:C\to C\) be a hemicontinuous and pseudocontractive mapping. Let \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) be the same as in Theorem 4.2. Then the sequence \(\{x_{n}\}\) generated by (4.5) converges in norm to \({x^{*}} = {P_{\operatorname{Fix}(T)}}u\).
Proof
5 Conclusion
We studied a class of monotone variational inequality problems (MVIPs) for generalized Lipschitz continuous, hemicontinuous, and monotone operators defined on a nonempty, closed, and convex subset of a real Hilbert space. Firstly, by using the maximal monotone theory, we established a new existence and uniqueness theorem for a variational inequality problem with a hemicontinuous and strongly monotone operator. Then, by virtue of the existence and uniqueness theorem, we introduced an implicit method and analyzed its strong convergence. We also introduced an explicit iterative method as discretization of the implicit method. We proved that both the implicit and the explicit methods converge in norm to the same solutions to the MVIPs. Finally, we applied our main results to the constrained minimization problems and the minimum-norm fixed point problems for generalized Lipschitz continuous, hemicontinuous, and pseudocontractive mappings.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (11071053).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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